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The Wikipedia page for the Tutte polynmomial doesn't use the word crossing, it also uses the word bridge. In graph theory, a bridge of a connected graph is an edge that separates the graph into two components.

However, there is a relation between the Tutte polynomial and the Jones and HOMFLY polynomials. More precisely, the HOMFLY polynomial generalizes the Tutte polynomial for planar graph. A knot diagram has crossings, which means points where two arcs of the knot cross. A knot diagram also has bridges; a bridge is a maximal sequence of over-crossings along an arc of the diagram. So there is a little bit of collision of terminology, because crossings aren't bridges and because bridges for knots aren't the same as bridges for graphs.